Shader turns black when connected to node group
This is because you are plugging a Shader output (green) into a color input (yellow). See What is the meaning of the color of the node sockets in the node editor?

Instead you should add a shader input (green) to the "Logo Overlay" group. Doing this is a little complicated, as you currently can't directly set the socket type without some python. However, it is possible:


Enter the node group (&#8633; Tab)
Plug the placeholder output of the Group Input node (hollow) into any input socket of the desired type. In this case, green for shader:



Optionally rename the new input in the properties panel (N).


Once this is done, your setup should work.



I don't recommend putting the material output node inside a node group, as this could lead to confusion down the road depending on how the group is used. Instead you might want to plug the mix shader into a Group Output node, and plug the entire group node into the material output in the main node tree.